Logic

Logic is the study of reasoning and valid inference. It involves analyzing statements, arguments, and deductive processes. Questions may include solving logic puzzles, evaluating the truth of compound statements, using truth tables, and identifying logical fallacies.

• Question

  The numbers `1`, `2`, `3`, ..., `8` are written on the vertices of a cube. Prove that there exists an edge of the cube such that the difference between the numbers at its endpoints is at least `3`.

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Combinatorics -> Pigeonhole Principle Combinatorics -> Graph Theory Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction

• Question

  The game takes place on an infinite plane. One player moves the wolf, and another player moves K sheep. After the wolf's move, one of the sheep makes a move, then the wolf again, and so on. In one move, the wolf or a sheep cannot move more than one meter in any direction. Can the wolf always catch at least one sheep, regardless of the initial positions?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Geometry -> Plane Geometry Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 2

• Question

  K friends simultaneously learn K pieces of news (one piece of news per friend). They begin to phone each other and exchange news. Each call lasts one hour. How long will it take for all friends to know all the news? Consider the cases:
  (a) (5 points) K=64
  (b) (10 points) K=55
  (c) (12 points) K=100
  (a) Answer

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 4

• A mistake in the exercise

  Prove that there is an error in the following multiplication problem:

  \(\begin{array}& & & * & * & * & 2 & 7 \\ \times & & & & & * & * \\ \hline & * & * & * & * & * & 6 \\ + & * & * & * & * & * & \\ \hline & * & * & * & * & 4 & 6 \end{array}\)

  Topics:

  Arithmetic Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 4

• Pine Trees in the Forest

  A forester counts pine trees in a forest. He walked along each of the circles in the image, and within each circle he counted exactly `3` pine trees. Prove that the forester surely made a mistake in his count.

  

  Topics:

  Logic -> Reasoning / Logic Set Theory Proof and Example -> Proof by Contradiction

• Question

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• Question

  From a chessboard, two opposite corners are removed (the squares `a1` and `h8`, for example). Can you tile the remaining board with dominoes?

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  Topics:

  Combinatorics -> Invariants Combinatorics -> Matchings Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring Puzzles and Rebuses

• Ants on a Stick

  On a stick that is one meter long, there are `10` ants, `5` from each side, at distances of one centimeter (see the picture). The ants on the left side of the stick move to the right, and the ants on the right side of the stick move to the left. The speed of each ant is constant and equal to one centimeter per second. When two ants meet, they both reverse direction and start moving away from each other. When any ant reaches the end of the stick, it falls off (ants are particularly stupid creatures).

  

  a. Will all the ants fall off the stick, and if so, in how much time?

  b. How many collisions will occur between the ants?

  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Bacteria in a Test Tube

  A scientist has a test tube containing bacteria. Every second, each bacterium splits into two. After two hours, the test tube was completely full of bacteria. How long before that was the test tube exactly half full?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards

• Question

  Find the sum of all natural numbers from `1` to `100`.

  Topics:

  Arithmetic Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence